# Full text of "Analytical Mechanics"

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```PERIODIC MOTION                            319
where & is a positive constant which depends upon the physical properties of the wire.* The negative sign indicates the fact that the torque and the angular displacement are oppositely directed. Substituting this value of G in the torque equation we have
/f = -ke,                  (xvii)
*--«•
k
where c2= -• But these are the typical forms of the equation of simple harmonic motion; therefore
(XVIII)
is the expression for the period. It will be observed that the motion is strictly harmonic; consequently there is no correction for finite amplitudes.
247. Application to the Determination of Moment of Inertia. — Let P be the period of the torsion pendulum and Pr its period after the body whose moment of inertia is desired is fastened to the bob of the pendulum. Further let I be the moment of inertia of the bob about the suspension wire as an axis and T the moment of inertia of the body. Then we have
P=2
and                          P' = 2
P'2_ p2
Therefore                    I'=     p2    I
i     A   «     I' and                                k = 4 IT* ~-T^—;
Hence if I is known both I' and k may be determined experimentally.
* Page 178.```